Then, by the Sobolev embedding and elliptic estimates (see [1]}), we obtain that
\(\Vert \chi _j^{(n)}(y,\cdot )\Vert _{L^\infty (0,1)}^2\lesssim \Vert \chi _j^{(n)}(y,\cdot )\Vert _{H_0^1 (0,1)}^2\le 1+\lambda _j[U^{(n)}(y,\cdot )] \lesssim j^2\) 
